Periodic harmonic functions on lattices and points count in positive characteristic

• Autorzy: Mikhail Zaidenberg
• Języki publikacji: EN

Abstrakty

EN

This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.

Słowa kluczowe

EN

Cellular automaton; Chebyshev-Dickson polynomial; Convolution operator; Lattice; Finite field; Discrete Fourier transform; Discrete harmonic function; Pluri-periodic function

Kategorie tematyczne

• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
• 11T99: None of the above, but in this section
• 11T06: Polynomials
• 31C05: Harmonic, subharmonic, superharmonic functions
• 43A99: None of the above, but in this section
• 37B15: Cellular automata

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2009
• Tom: 7
• Numer: 3
• Strony: 365-381

Daty

wydano

2009-09-01

online

2009-08-12

Twórcy

autor

Mikhail Zaidenberg

• Université Grenoble I

Bibliografia

• [1] Amin A.T., Slater P.J., Zhang G.-H., Parity dimension for graphs-a linear algebraic approach, Linear and Multilinear Algebra, 2002, 50, 327–342 http://dx.doi.org/10.1080/0308108021000049293
• [2] Barua R., Ramakrishnan S., σ-game, σ+-game and two-dimensional additive cellular automata, Theoret. Comput. Sci., 1996, 154, 349–366 http://dx.doi.org/10.1016/0304-3975(95)00091-7
• [3] Bhargava M., Zieve M.E., Factoring Dickson polynomials over finite fields, Finite Fields Appl., 1999, 5, 103–111 http://dx.doi.org/10.1006/ffta.1998.0221
• [4] Bicknell M., A primer for the Fibonacci numbers VII, Fibonacci Quart., 1970, 8, 407–420
• [5] Goldwasser J., Klostermeyer W., Ware H., Fibonacci Polynomials and Parity Domination in Grid Graphs, Graphs Combin., 2002, 18, 271–283 http://dx.doi.org/10.1007/s003730200020
• [6] Goldwasser J., Wang X., Wu Y., Does the lit-only restriction make any difference for the σ-game and σ+-game?, European J. Combin., 2009, 30, 774–787 http://dx.doi.org/10.1016/j.ejc.2008.09.020
• [7] Gravier S., Mhalla M., Tannier E., On a modular domination game, Theoret. Comput. Sci., 2003, 306, 291–303 http://dx.doi.org/10.1016/S0304-3975(03)00285-8
• [8] Heath-Brown D.R., Artin’s conjecture for primitive roots, Quart. J. Math., 1986, 37, 27–38 http://dx.doi.org/10.1093/qmath/37.1.27
• [9] Hoggatt V.E.Jr., Bicknell-Johnson M., Divisibility properties of polynomials in Pascal’s triangle, Fibonacci Quart., 1978, 16, 501–513
• [10] Hoggatt V.E.Jr., Long C.T., Divisibility properties of generalized Fibonacci polynomials, Fibonacci Quart., 1974, 12, 113–120
• [11] Humphreys J.E., Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1978
• [12] Hunziker M., Machiavelo A., Park J., Chebyshev polynomials over finite fields and reversibility of σ-automata on square grids, Theoret. Comput. Sci., 2004, 320, 465–483 http://dx.doi.org/10.1016/j.tcs.2004.03.031
• [13] Jacob G., Reutenauer C., Sakarovitch J., On a divisibility property of Fibonacci polynomials, preprint available at http://en.scientificcommons.org/43936584
• [14] Levy D., The irreducible factorization of Fibonacci polynomials over Q, Fibonacci Quart., 2001, 39, 309–319
• [15] Lidl R., Mullen G.L., Turnwald G., Dickson polynomials, Longman Scientific and Technical, Harlow, John Wiley and Sons, Inc., New York, 1993
• [16] Martin O., Odlyzko A.M., Wolfram S., Algebraic properties of cellular automata, Comm. Math. Phys., 1984, 93, 219–258 http://dx.doi.org/10.1007/BF01223745
• [17] Moree P., Artin’s primitive root conjecture-a survey, preprint available at http://arxiv.org/abs/math/0412262
• [18] Ram Murty M., Artin’s conjecture for primitive roots, Math. Intelligencer, 1988, 10, 59–67 http://dx.doi.org/10.1007/BF03023749
• [19] Sarkar P., Barua R., Multidimensional σ-automata, π-polynomials and generalised S-matrices, Theoret. Comput. Sci., 1998, 197, 111–138 http://dx.doi.org/10.1016/S0304-3975(97)00160-6
• [20] Sutner K., σ-automata and Chebyshev-polynomials, Theoret. Comput. Sci., 2000, 230, 49–73 http://dx.doi.org/10.1016/S0304-3975(97)00242-9
• [21] Webb W.A., Parberry E.A., Divisibility properties of Fibonacci polynomials, Fibonacci Quart., 1969, 7, 457–463
• [22] Zaidenberg M., Periodic binary harmonic functions on lattices, Adv. in Appl. Math., 2008, 40, 225–265 http://dx.doi.org/10.1016/j.aam.2007.01.004
• [23] Zaidenberg M., Convolution equations on lattices: periodic solutions with values in a prime characteristic field, In: Kapranov M., Kolyada S., Manin Y.I., Moree P., Potyagailo L.A. (Eds.), Geometry and Dynamics of Groups and Spaces, In Memory of Alexander Reznikov, Progress in Mathematics 265, 719–740, Birkhäuser, 2008

Identyfikatory

DOI

10.2478/s11533-009-0029-0